Question

Write each expression as a sum and/or difference of logarithms. Express powers as factors. $$ \log _{2}\left(\frac{x^{3}}{x-3}\right) \quad x>3 $$

Short answer

\(3 \log_{2}(x) - \log_{2}(x-3)\)

Step-by-step solution

Apply the Quotient Rule of Logarithms

The first step is to apply the quotient rule of logarithms, which states that \ \ \ \ \ \ \ \ \ \ \log_a(\frac{M}{N}) = \log_a(M) - \log_a(N).\ So, apply it to the given expression:\ \ \ \(\log_{2}\left(\frac{x^{3}}{x-3}\right) = \log_{2}(x^{3}) - \log_{2}(x-3)\)

Apply the Power Rule of Logarithms

Next, use the power rule of logarithms, which states that \ \ \ \ \ \ \ \ \ \ \log_a(M^p) = p\log_a(M). \Apply this rule to \log_{2}(x^{3}): \ \ \ \(\log_{2}(x^{3}) = 3 \log_{2}(x)\)

Combine the Results

Now, combine the results from steps 1 and 2 to write the original logarithmic expression as a sum and/or difference of logarithms with powers expressed as factors:\ \ \ \(\log_{2}\left(\frac{x^{3}}{x-3}\right) = 3 \log_{2}(x) - \log_{2}(x-3)\)

Key concepts

quotient rule of logarithms

The quotient rule of logarithms is an essential tool when dealing with divisions inside a logarithm. It says that the logarithm of a quotient is equal to the difference of the logarithms. Mathematically, we write this as:

\[ \log_a \left(\frac{M}{N}\right) = \log_a (M) - \log_a (N) \]

This means that if you have a logarithmic expression involving a division, you can split it into two separate logarithms, one for the numerator and one for the denominator.

Let's see this with an example. Consider the expression \( \log_{2} \left(\frac{x^3}{x-3}\right) \). Using the quotient rule, we can rewrite it as:

\[ \log_{2} (x^3) - \log_{2} (x-3) \]

This simplifies our original expression by separating it into two manageable parts. The next step involves dealing with the logarithm of the numerator.

power rule of logarithms

The power rule of logarithms allows us to handle exponents within a logarithm. It states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the base number. Mathematically, it looks like this:

\[ \log_a (M^p) = p \log_a (M) \]

So, if we have a logarithmic expression where the number is raised to an exponent, we can bring the exponent to the front. This helps us to simplify the expression.

Let's apply this to our previous example component \( \log_{2} (x^3) \). Using the power rule, it becomes:

\[ \log_{2} (x^3) = 3 \log_{2} (x) \]

Here, we moved the exponent 3 in front of the logarithm, simplifying it. This makes it easier to handle, collect terms, and solve.

sum and difference of logarithms

When working with logarithms, expressing them as a sum or difference is often very useful. Using the rules of logarithms, we can break down more complicated expressions.

Recall from the quotient rule and power rule:

• Quotient rule: \( \log_a \left(\frac{M}{N}\right) = \log_a (M) - \log_a (N) \)
• Power rule: \( \log_a (M^p) = p \log_a (M) \)

Now, putting it all together, let's combine our results into one final, easy-to-manage expression. Starting with:

\[ \log_{2} \left(\frac{x^3}{x-3}\right) \]

We used the quotient rule to break it down:

\[ \log_{2} (x^3) - \log_{2} (x-3) \]

Then, applying the power rule on the first term, we get:

\[ 3 \log_{2} (x) - \log_{2} (x-3) \]

This simplification using sums and differences of logarithms makes the expressions easier and often more intuitive to work with. It’s particularly helpful for solving equations and analyzing complex logarithmic functions.